Trajectory optimization algorithm for a constant altitude cruise flight ...

Trajectory optimization algorithm for a constant altitude cruise flight with a required time of arrival constraints Alexandre Liv1, Radu Ioan Dancila2 and Ruxandra Mihaela Botez3 ÉTS, Laboratoire de recherche en commande active, avionique et en aéroservoélasticité (LARCASE) Montreal, Quebec, H3C 1K3, Canada

Abstract Decreasing the flight costs, therefore flying along the optimal for a trajectory, is a constant preoccupation for all aircraft operators. Moreover, the reduction of fuel consumption and pollutant emissions is a significant factor in the trajectory optimization analysis. The study presented in this article is part of the research conducted at the Laboratoire de recherche en commande active, avionique et aéroservoélasticité (LARCASE), at École de Technologie Supérieure, in the field of aircraft trajectory optimization algorithms for Flight Management System platforms, and investigates an optimization algorithm for a leveled cruise flight segment with required time of arrival constraint. The proposed algorithm is deterministic, in accordance with the requirements for avionics equipment.

Nomenclature CI EXH FB FMS GW LNAV Opt RTA VNAV

= Cost Index = Exhaustive search = Fuel Burn = Flight Management System = Gross Weight = Lateral Navigation = Optimization = Required Time of Arrival = Vertical Navigation

Introduction The study is part of the ongoing work conducted at the Laboratoire de recherche en commande active, avionique et aéroservoélasticité (LARCASE) in the field of green aircraft trajectories optimization. This project is in collaboration with CMC Electronics – Esterline and is part of the new excellence center network directed by the Green Aviation Research Network (GARDN) funded by the Canadian Government [1]. The FMS (Flight Management System) platform is in a constant evolution, it is continuously updated in order to enhance it [2]. The aim of the research is to conceive and analyze new strategies and 1

Undergraduate Internship Student, LARCASE, 1100 Notre-Dame Ouest, Montréal, QC H3C 1K3, Canada Ph.D. Student, LARCASE, 1100 Notre-Dame Ouest, Montréal, QC H3C 1K3, Canada 3 Professor, LARCASE, 1100 Notre-Dame Ouest, Montréal, QC H3C 1K3, Canada 2

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methods for lateral and vertical flight trajectory optimization in order to reduce the fuel consumption and pollutant emissions and, then, to implement efficient flight trajectories optimization algorithms that would improve the FMS (Flight Management System) platform’s performance and capabilities. Many researches have been conducted at the LARCASE in the field of flight trajectory optimization. The fuel burn is an important parameter to take into account in the optimization. An algorithm to predict the quantity of fuel burned by an aircraft flying at constant speed and constant altitude has been developed for use by the FMS [3]. Moreover, flying with less pollutant emission is a current issue for aeronautics companies. New methods have been implemented to estimate the fuel burn and emissions generated (such as CO2 or NOx) during missed approach procedure [4] [5]. Vertical navigation trajectory optimization is a means to reduce fuel consumption. Algorithms implemented at the laboratory optimize the speeds and altitudes for the vertical profile, obtaining a trajectory that reduces the global flight cost [6] [7]. Many approaches to optimize predictions of the FMS, regarding the VNAV, have been studied at the LARCASE. Methods based on aircraft performance database using linear interpolation with the help of the Lagrange method have been used for implementation [8] [9]. Other approaches have been conducted such as the use of genetic algorithms [10]. To find the optimal path in 4D (3 space dimensions and 1 time dimension) minimizing the total cost of a flight, the LARCASE also works on dynamic programming [11], and a method consisting in connecting all the waypoints specified in the flight plan of an aircraft and including the management of No-Fly-Zones (zones over which flying is forbidden or temporarily refused) [12]. A flight trajectory is composed of three main phases: a climb, a cruise and a descent. This paper investigates the speed optimization for a flight trajectory in cruise, at constant altitude. The cruise is the longest phase of a flight and, consequently, an interesting segment for fuel and cost optimization [13]. The segment under optimization is considered to be formed by a succession of sub-segments, each one delimited by two adjacent waypoints of the LNAV (Lateral Navigation) plan and flown at constant Mach Speed. The optimization objective is to identify the set of sub-segment cruise speeds that minimizes the total cost for the flight and results in a flight time within the imposed RTA (Required Time of Arrival) constraint limits. Finding the optimal flight profile by evaluating all the possible segment speed combinations (segment Mach speed values) is not a feasible solution due to the high number of flight profile evaluations required and the limited computing power of the FMS platforms. The objective of this investigation is to find a method that minimizes the amount of flight profile evaluations necessary to identify the optimal trajectory (the set of segment speed values that yield the minimum total cost). However, a compromise between the optimality of the solution and the program execution time has to be taken into consideration. The proposed algorithm is deterministic. The optimal set of segment speeds search starts from the set economic segment speeds, defined in [14], determined using the cost characteristic for each subsegment. Then, the optimal profile is found through iterative analysis of the global flight cost variation/flight time variation ratio of multiple speed changes for individual sub-segments.

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Methodology The optimization algorithm presented in this study has been implemented in MATLAB and evaluated using the flight performance data of a transportation aircraft model. The aircraft flight performance data is based on linear interpolation tables, a model used by the FMS platforms. The set of hypotheses and assumptions used in this study are: - The airplane flies at constant altitude along the cruise trajectory under optimization; - The set of speed values considered for the optimization are Mach values. Each segment speed can take any value within the imposed speed restrictions; - The accelerations and decelerations are not taken into account; - The wind value is constant for each sub-segment composing the LNAV plan; - The turn segments between two sub-segments with different headings are not included, the heading change is considered immediate; - During the search of the optimal profile, the algorithm should evaluate the least number of segment speed sets; - It should use the least memory space. The aim of the research is to identify and evaluate a deterministic algorithm that finds the optimal segment speeds profile which complies with a RTA constraint and yields a minimal total cost, while requiring the smallest amount of profile evaluations. Given a flight altitude and aircraft configuration, the optimization algorithm finds the flight at economic speed on each sub-segment and evaluates various speed changes for individual subsegments in order to obtain the required flight time and the optimal speed combination. To validate the optimization algorithm results, the optimal set of segment speeds determined by an exhaustive search is used as reference. The exhaustive search evaluates every possible segment speeds combinations, where each segment speed value varies with a 0.001 Mach step, while taking into consideration the speed restrictions (valid speed domain) for each sub-segment. For this step of the search, the study is focused on two sub-segments and three sub-segments scenarios. The total flight cost value, the set of segment speeds and the total flight time of the optimal profile found by the optimization algorithm and those found by the exhaustive search are compared and the number of evaluations is calculated.

Algorithm The optimization algorithm can be divided into two main parts. The first one searches the combination of economic speeds without taking into account the time constraint (the starting point of the optimal speed profile). Then, starting from this initial point, the RTA is taken into consideration and the optimal profile is found by analyzing total flight time and total cost variations. I.

The economic speeds combination

An economic speed is the minimal cost speed of a single sub-segment [15]. For each sub-segment, the minimal cost speed is found with the help of a minimum value search. Indeed, the variation of the total cost function of the flight speed follows a curve that has a unique minimum [16]. The cost characteristic, and consequently the economic speed for each sub-segment, depends on the Cost Index (CI) (the time cost/fuel cost ratio) [17]. When the CI is equal to 0, the flight time costs 3

are taken into consideration, so the fuel burn is minimized. If the CI is at its maximum value, the profile with a minimal flight time is preferred [17]. At the end of this part, the set of economic segment speeds is found and considered as the starting point for the optimal profile search. This speed set could be the one that returns the global minimum total cost. This section doesn’t take into account of time constraints. The following paragraphs will explain the methodology used to determine the economic speed for one subsegment of the LNAV plan.  Initialization The aircraft speed on the analyzed sub-segment can take one of the values included between the imposed speed restrictions Mmin and Mmax, in Mach. An initial increment/decrement step is set with a value equal to 20% of the speeds range as: (1) For each sub-segment, the initial gross weight is the final gross weight of the previous sub-segment. The first segment's gross weight takes the value imposed by the given optimization scenario. To start the minimal cost speed search, an initial point is found, its speed value is the “middle” one between the restrictions, equal to .

Figure 1. Starting point of the search

 The minimum cost search method Starting from the point found previously, while the minimum cost speed isn’t found, the following steps are executed: The “middle point” speed total cost is calculated as well as the cost for two other speeds – on the right and on the left, according to the increment/decrement step.

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Figure 2. Evaluation

The total costs corresponding to the three evaluated speeds are compared. The result of these comparisons can fall into one of the following three situations:  If the decrement speed cost and the increment speed cost are higher than the middle one, and the step value is at its minimal threshold given at the beginning of the algorithm, then the search is interrupted because the program considers that the economic speed is found.  Among the comparison speeds (increment and decrement points), if no speed cost is lower than the middle one and the increment/decrement step is not at its minimal threshold, then the step is cut by five and the search continues with this new step value, keeping the same reference speed “middle”.  If at least one of the two comparison speeds has a lower cost than the middle one, then the point is kept and becomes the new reference speed – the new middle, for the next iteration. In addition, the increment/decrement step is reduced.

Figure 3. New reference speed for the next iteration

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Figure 4. Increment/decrement step reduction for the next iteration

The program keeps running until the minimum cost speed is found: when the evaluation step is at its minimal value and the right and left speeds have higher costs. Once the economic speed is found for the first sub-segment, the fuel burn (FB) is calculated and the final aircraft gross weight is calculated [18], as follows: For the i-th sub-segment, (2) The next segment gross weight takes the value of the final GW. (3) This algorithm is applied for each sub-segment of the LNAV plan, and returns the economic speeds combination at the end of the program. This profile is the starting point for the second section of the optimization algorithm.  Results of the first part of the algorithm In order to know if the economic speeds combination returns the global minimal cost, results from an exhaustive search are used. For the algorithm validation, 62 tests have been done with the EXH (Exhaustive search) and the minimum search algorithms, for two-sub-segments scenarios and three-sub-segments scenarios. Sixty two combinations of CI and RTA values have been evaluated. Each evaluation returns the speed set, the flight total cost, the number of evaluated profiles and the execution time. The global minimum cost profile is the speed set, specific to a scenario, yielding the lowest cost without taking account of the RTA constraint. Each test has shown that the set of segments flying at economic speeds corresponds to the global minimum cost profile. Only one three-sub-segments scenario result is presented in this paper.

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Table 1. Three-sub-segments scenario results

CI 0 50 100 200 225 300 475 600 999

Economic speeds set (Mach) [0.7800 0.7800 0.7800] [0.7800 0.7800 0.7800] [0.7800 0.7800 0.7800] [0.8000 0.8000 0.8000] [0.8000 0.8200 0.8200] [0.8200 0.8200 0.8200] [0.8200 0.8200 0.8400] [0.8400 0.8400 0.8400] [0.8400 0.8400 0.8400]

Execution time 4.9303 s 4.8908 s 4.9293 s 5.5042 s 7.7967 s 5.5692 s 11.4868 s 5.0191 s 5.044 s

Global minimum cost speeds set (Mach) [0.7800 0.7800 0.7800] [0.7800 0.7800 0.7800] [0.7800 0.7800 0.7800] [0.8000 0.8000 0.8000] [0.8000 0.8200 0.8200] [0.8200 0.8200 0.8200] [0.8200 0.8200 0.8400] [0.8400 0.8400 0.8400] [0.8400 0.8400 0.8400]

Validation ? Yes Yes Yes Yes Yes Yes Yes Yes Yes

The first section of the optimization algorithm finds the global minimum cost speeds set with a low execution time (5 to 11 seconds).

II.

Finding the optimal profile

The objective of the second part of the optimization algorithm is to find the combination of subsegment speeds minimizing the total flight cost and resulting in a flight time compliant with the RTA, using the economic speeds profile as a starting point. It implies that the time constraints have to be taken into account. Before starting the search, the global minimum cost is evaluated to know if the profile is “RTA compliant”. Indeed, if the speeds combination yields a minimal total cost and complies with the imposed flight time, then it is considered as the optimal one and the algorithm is stopped.  The algorithm The method consists in a variable step search on each sub-segment when the speeds are changed for individual sub-segments. Therefore, an initial constant evaluation step is assigned to each partition of the LNAV plan. The first reference profile is the one found previously in the first part of the program. Depending on the value of its total flight time Tref, three possible cases are presented:  

If Tref < RTA, the reference combination speeds are reduced to obtain RTA compliant profiles; Else, if Tref > RTA, the reference combination speeds are increased;



Else, if Tref = RTA, as explained previously, the reference profile is the optimal one.

Thus, the gradient of the segment speed to time variation can be evaluated in order to estimate the segment speed change that results in a flight time closer to the RTA constraint. Consequently, from the beginning of the algorithm, the way of speeds variation segment by segment is known. Indeed, after having evaluated the economic speeds set's flight time and compared it to the RTA value, the optimal profile search follows the direction of the flight time increment or decrement - depending on the comparison between the ECON flight time and the RTA. As an example, a scenario with a flight segment divided into two sub-segments with specific speed restrictions (Mach 0.78 to Mach 0.84 for each segment) is used. 7

”RTA zone” including the optimal profile First reference profile

Figure 5. Direction of the search

This figure illustrates a 2-dimensions graph, each axis corresponds to one sub-segment speed values. A dot is a combination of two speeds. The first reference profile corresponds to the starting point, the economic speeds flight profile. The speed combinations resulting in a flight time in accordance with the RTA are represented in the black area, in which the optimal profile is located. By comparison with the RTA, the direction of the optimal speeds combination search is known starting from the economic trajectory. The optimization algorithm, knowing the direction of search to reach the required time of arrival, evaluates iteratively flight profiles with various speed changes for individual sub-segments. When a speed is incremented or decremented (according to the direction of search), a comparison profile is created. At each step of the evaluation, there are n comparison speed sets, n being the total number of sub-segments composing the LNAV plan. Each comparison profile is obtained for each speed increment or decrement segment. Among the comparison profiles, if one of them complies with the required time of arrival and if the evaluation step on the modified speed sub-segment is higher than the minimal step threshold – constant during all the execution, the program reduces the increment/decrement step on this subsegment, and the operation keeps iterating with the same reference profile. However, if the obtained comparison profile flight time is included in the time constraint interval and if the evaluation step on the modified speed sub-segment is at its minimum, then the program considers it as the optimal profile and the optimization algorithm is stopped. The global optimal speed set is not necessarily found by the optimization algorithm, but a similar one with a bit higher flight cost, because the optimization search is starting from the economic speeds set and the speed variation, in order to find the optimal profile, takes into account the cost and flight time variations. Else, if none of these two conditions is confirmed, a cost variation/time variation method is used. At the iteration, the total flight cost gradient and the total flight time gradient are calculated between the reference profile and the comparison ones.

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The goal of this function is to get the lowest variation of cost compared to the highest time variation. Indeed, to quickly reach a modified speeds combination complying with the RTA, a high variation of total flight time is needed. Moreover, to get the optimal profile, a low variation of the total cost is preferred. To this end, a cost/time ratio is established, as follows: |

|

(4)

The algorithm aim is to select the comparison profile, among the n calculated ones, minimizing this ratio called “g”. This will be the new reference profile for the next program iteration. All these steps keep running until the algorithm returns the optimal profile, RTA compliant for a minimal cost. For instance using a two-sub-segments scenario, the search direction is represented in a graph (Figure 6). At each iteration, there are two comparison profiles pour a two-sub-segments scenario. The one with the lowest “g” ratio is selected and becomes the next reference dot. The direction to the optimal profile follows the bold arrows.

Figure 6. Representation of a full search, starting from the economic speeds profile for a two-sub-segments scenario

Results Various CI and RTA value combinations for three two-sub-segments scenarios and three three-subsegments scenarios are created. Then, the segment speeds combinations, the corresponding total flight costs; the number of evaluated profiles and execution time obtained by the optimization algorithm are presented. For each test scenario, the results of the exhaustive search algorithm: the speed profile, total cost and number of evaluated profiles data, are used as a reference. To analyze the results given by the two programs, a comparison table is created, using: - The percent relative error according to this formula: (5)

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If the percentage is lower than 0, this means that the speed combination found by the optimization function returns a lower cost than the exhaustive search. Else, if the percentage is higher than zero, the segment speeds set returned by the optimization algorithm yields a higher cost. Finally, a null percentage indicates a same cost returned by the two algorithms. -

The percent ratio of the number of profiles evaluated by the optimization algorithm relative to the exhaustive search: (6)

This section of the article will show the two-sub-segments results and one three-sub-segment scenario results. Two-sub-segments results comparison Table 2. Two-sub-segments scenarios results obtained by the optimization algorithm

Scenario

1 (short1)

2 (short3)

3 (short6)

Total cost (kg)

Number of evaluated profiles

Execution time (s)

CI

RTA (s)

Opt algorithm speed set (Mach)

0

2 970

[0.8106, 0.8200]

4 097.516

37

11.0681

300

3 000

[0.8105, 0.8200]

19 022.298

27

9.4506

600

2 900

[0.8400, 0.8400]

33 678.392

3

4.1429

999

2 950

[0.8264, 0.8397]

53 261.792

29

9.5192

150

2 850

[0.8200, 0.8030]

11 104.796

31

10.2941

400

2 800

[0.8379, 0.8200]

23 023.061

29

10.3378

720

2 900

[0.8106, 0.7999]

38 490.783

41

11.879

850

2 790

[0.8400, 0.8400]

43 985.019

3

4.1399

0

1 260

[0.7800, 0.7800]

1 558.849

3

2.1622

50

1 200

[0.8200, 0.8138]

2 697.194

37

5.685

150

1 258

[0.7997, 0.7920]

4 706.216

29

5.221

400

1 190

[0.8394, 0.8200]

9 765.213

27

5.1873

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Table 3. Two-sub-segments scenarios results obtained by the exhaustive search

Scenario

CI

1 (short1)

0 300 600 999 150 400 720 850 0 50 150 400

2 (short3)

3 (short6)

RTA (s) EXH speed set (Mach) Total cost (kg) 2 970 3 000 2 900 2 950 2 850 2 800 2 900 2 790 1 260 1 200 1 258 1 190

[0.8150, 0.8030] [0.8110, 0.8180] [0.8400, 0.8400] [0.8270, 0.8370] [0.8200, 0.8030] [0.8380, 0.8200] [0.8080, 0.8040] [0.8400, 0.8400] [0.7800, 0.7800] [0.8190, 0.8130] [0.8000, 0.7920] [0.8390, 0.8200]

4 099.016 19 022.53 33 678.392 53 263.94 11 104.84 23 023.23 38 492.06 43 985.019 1 558.849 2 697.208 4 706.2 9 765.174

Number of evaluated profiles

3 721

Table 4. Opt/EXH algorithm comparison

Scenario 1 (short1)

2 (short 3)

3 (short 6)

CI 0 300 600 999 150 400 720 850

RTA (s) 2970 3000 2900 2950 2850 2800 2900 2790

Cost relative error (%) -0.0366 % -0.0012 % 0% -0.0040 % -0.0004 % -0.0007 % -0.0033 % 0%

Percent evaluations ratio (%) 0.994 % 0.725 % 0.080 % 0.779 % 0.833 % 0.779 % 1.101 % 0.080 %

0 50 150 400

1260 1200 1258 1190

0% -0.0005 % +0.0003 % +0.0004 %

0.080 % 0.994 % 0.779 % 0.725 %

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Three-sub-segments scenario results comparison The same study has been done on a three-sub-segments LNAV trajectory: Table 5. Three-sub-segments scenario results obtained by the optimization algorithm

Scenario

3segm1

CI

RTA (s)

Opt algorithm speed set (Mach)

Total cost (kg)

Number of evaluated profiles

Execution time (s)

0

4 020

[0.8230, 0.8400, 0.8400]

6 266.557

60

22.2766

200

4 100

[0.8042, 0.8200, 0.8200]

19 424.386

51

19.8596

400

4 200

[0.7964, 0.8000, 0.8000]

33 357.155

54

21.1589

600

4 150

[0.8000, 0.8136, 0.8200]

46 984.352

60

21.7894

750

4 180

[0.8000, 0.8014, 0.8197]

57 587.465

69

24.4441

999

4 090

[0.8197, 0.8244, 0.8400]

73 797.256

45

17.8607

Table 6. Three-sub-segments scenario results obtained by the exhaustive search

Scenario

3segm1

CI

RTA (s)

EXH algorithm speed set (Mach)

Total cost (kg)

0

4 020

[0.8230, 0.8400, 0.8400]

6 266.557

200

4 100

[0.8050, 0.8200, 0.8190]

19 424.62

400

4 200

[0.7970, 0.8000, 0.7990]

33 357.03

600

4 150

[0.8040, 0.8130, 0.8150]

46 984.66

750

4 180

[0.8040, 0.8000, 0.8180]

57 587.461

999

4 090

[0.8210, 0.8240, 0.8390]

73 796.52

Number of evaluated profiles

226 981

Table 7. Opt/EXH algorithms comparisons

Scenario

CI

RTA (s)

Cost percent error (%)

Percent evaluations ratio (%)

0

4 020

0%

0.03 %

200

4 100

-0.001 %

0.02 %

400

4 200

+0.0003 %

0.02 %

600

4 150

-0.0007 %

0.03 %

750

4 180

+0.000007 %

0.03 %

999

4 090

+0.001 %

0.02 %

3segm1

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Discussion Analyzing the results obtained using the two and three-sub-segments test scenarios we noticed that the speed sets obtained by the optimization algorithm are different from the ones found by exhaustive search. However, we can observe that the Mach speed values are quite similar. The cost differences given by the comparison tables present a low percentage (from +0.0003% to +0.0004% for the two-sub-segments trajectories and +0.000007% to +0.001% for the three-subsegments one); several tests have identical cost values. Some cost relative errors have negative values; indeed, the optimization program yields a lower cost than the exhaustive algorithm for theses test cases. Finally, according to the comparison tables for the evaluated test scenarios, the proposed optimization algorithm evaluates a lower number of speed profiles until it reaches the optimal speed set. Actually, for the two-sub-segments tests, the program evaluates between 0.08% and 1.1% of the possible speed combinations. In the case of the three-sub-segments, it evaluates between 0.02% and 0.03% of the total number of profiles evaluated by the exhaustive search. To conclude, the implemented optimization algorithm is considered to comply with the initial objective of the project: finding the optimal segment speeds combination satisfying the required time of arrival initially set, for a minimum total flight cost and a reasonable execution time. However, the optimization program doesn’t find “the” optimal profile found by the exhaustive search algorithm, but a close one, and sometimes better, in a limited number of evaluated profiles. A compromise between the trajectory “optimality level” and the amount of evaluations had to be considered during the implementation.

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